There is free entry to the Vatican Museums on the last Sunday of every month, between 09.00 and 12.30, leaving by 14.00.
It is not possible to reserve tickets, they must be bought that day at the museum ticket office.
Aside from the last Sunday of each month, the Vatican Museums are not open on Sundays.
The initiative is held each month provided the date does not coincide with Easter Sunday, the Feast of Sts Peter and Paul (29 June), Christmas Day (25 December), and St Stephen's Day (26 December).
For ticket details see Vatican website.